Section SET Sets

Definition SETSet A set is an unordered collection of objects. If
S is a set and
x is an object that
is in the set S,
we write x ∈ S.
If x is not in
S, then we
write x∉S.
We refer to the objects in a set as its elements.

Hard to get much more basic than that. Notice that the objects in a set can be
anything, and there is no notion of order among the elements of the set. A set can
be finite as well as infinite. A set can contain other sets as its objects. At
a primitive level, a set is just a way to break up some class of objects
into two groupings: those objects in the set, and those objects not in the
set.

Example SETMSet membership From the set of all possible symbols, construct the following set of three
symbols,

Then the statement ■∈ S is true,
while the statement ▴∈ S is false.
However, then the statement ▴∉S
is true. ⊠

A portion of a set is known as a subset. Notice how the following
definition uses an implication (if whenever…then…). Note too how the
definition of a subset relies on the definition of a set through the idea of set
membership.

Definition SSETSubsetIf S and
T are two
sets, then S is
a subset of T,
written S ⊆ T if
whenever x ∈ S
then x ∈ T.

What does it mean for two sets to be equal? They must be the same.
Well, that explanation is not really too helpful, is it? How about: If
A ⊆ B and
B ⊆ A, then
A equals
B. This gives us something
to work with, if A
is a subset of B,
and vice versa, then they must really be the same set. We will now make the symbol
“ =”
do double-duty and extend its use to statements like
A = B, where
A and
B are
sets. Here’s the definition, which we will reference often.

Definition SESet Equality Two sets, S
and T, are
equal, if S ⊆ T and
T ⊆ S. In this case,
we write S = T.

Sets are typically written inside of braces, as
\left \{\ \right \}, as
we have seen above. However, when sets have more than a few elements, a
description will typically have two components. The first is a description of the
general type of objects contained in a set, while the second is some sort of
restriction on the properties the objects have. Every object in the set must be of
the type described in the first part and it must satisfy the restrictions in the
second part. Conversely, any object of the proper type for the first part, that
also meets the conditions of the second part, will be in the set. These
two parts are set off from each other somehow, often with a vertical bar
(\vert ) or a
colon (:).

I like to think of sets as clubs. The first part is some description of the type of
people who might belong to the club, the basic objects. For example, a bicycle
club would describe its members as being people who like to ride bicycles. The
second part is like a membership committee, it restricts the people who are
allowed in the club. Continuing with our bicycle club analogy, we might decide
to limit ourselves to “serious” riders and only have members who can
document having ridden 100 kilometers or more in a single day at least one
time.

The restrictions on membership can migrate around some between the first
and second part, and there may be several ways to describe the same set of
objects. Here’s a more mathematical example, employing the set of all integers,
ℤ, to
describe the set of even integers.

Notice how this set tells us that its objects are integer numbers (not, say, matrices
or functions, for example) and just those that are even. So we can write that
10 ∈ E,
while 17∉E
once we check the membership criteria. We also recognize the question

The union and intersection of sets are operations that begin with two sets and
produce a third, new, set. Our final operation is the set complement,
which we usually think of as an operation that takes a single set and
creates a second, new, set. However, if you study the definition carefully,
you will see that it needs to be computed relative to some “universal”
set.

Definition SCSet Complement Suppose S is a set that is a
subset of a universal set U.
Then the complement of S,
denoted \overline{S},
is the set whose elements are those that are elements of
U and not
elements of S.
More formally,

\eqalignno{
x ∈\overline{S}\text{ if and only if }x ∈ U\text{ and }x∉S & &
}

Notice that there is nothing at all special about the universal set. This is simply a term that
suggests that U
contains all of the possible objects we are considering. Often this set will be clear
from the context, and we won’t think much about it, nor reference it in our
notation. In other cases (rarely in our work in this course) the exact nature of the
universal set must be made explicit, and reference to it will possibly be carried
through in our choice of notation.

There are many more natural operations that can be performed on sets, such
as an exclusive-or and the symmetric difference. Many of these can be defined in
terms of the union, intersection and complement. We will not have much need of
them in this course, and so we will not give precise descriptions here in this
preliminary section.

There is also an interesting variety of basic results that describe the interplay
of these operations with each other. We mention just two as an example, these are
known as DeMorgan’s Laws.

Besides having an appealing symmetry, we mention these two facts, since
constructing the proofs of each is a useful exercise that will require a solid
understanding of all but one of the definitions presented in this section. Give it a
try.